Lattices of choice functions and consensus problems
Bernard Monjardet, Raderanirina Vololonirina
To cite this version:
Bernard Monjardet, Raderanirina Vololonirina. Lattices of choice functions and consensus
problems. Social Choice and Welfare, Springer Verlag, 2004, 23, pp.349-382.
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Lattices of choice functions and consensus problems
Bernard Monjardet
∗
Vololonirina Raderanirina
†
Abstract
In this paper we consider the three classes of choice functions satisfying the three significant
axioms called heredity (H), concordance (C) and outcast (O). We show that the set of choice
functions satisfying any one of these axioms is a lattice, and we study the properties of these
lattices. The lattice of choice functions satisfying (H) is distributive, whereas the lattice of
choice functions verifying (C) is atomistic and lower bounded, and so has many properties.
On the contrary, the lattice of choice functions satisfying (O) is not even ranked. Then using
results of the axiomatic and metric latticial theories of consensus as well as the properties
of our three lattices of choice functions, we get results to aggregate profiles of such choice
functions into one (or several) collective choice function(s).
Key-words: Choice function, lattice, concordance, heredity, outcast, Sperner family, median, consensus.
Title running: Lattices of choice functions
Proofs send to Monjardet B
JEL classification numbers: D71
AMS classification numbers: 91 B14
∗
CERMSEM, Université Paris 1, 106–112 boulevard de l’Hopital, 75647 Paris Cedex 13, France.
Email : [email protected]
†
CERMSEM, Université Paris 1, 106–112 boulevard de l’Hopital, 75647 Paris Cedex 13, France.
Email : [email protected]
1
1
Introduction
Arrow’s theorem has fostered a lot of work on the problem to aggregate individual preferences into a collective preference or more generally into a collective choice function. Still
more generally the ”russian school” (see e.g. [1], [2] or [3]) has considered the problem to
aggregate individual choice functions into a collective choice function. In their work they
emphasize the role of three axioms on choice functions, namely the heredity (or α or Chernoff
or heritage) axiom (H), the concordance (or γ or expansion) axiom (C) and the Outcast (or
Nash) axiom (O). Indeed, the combination of these axioms gives significant classes of choice
functions. For instance, a choice function is ”classically” rational (respectively rationalizable by a partial order, or path-independent) if and only if it satisfies axioms (H) and (C)
(respectively axioms (H),(C) and (O), or axioms (H) and (O)). Moreover, one can describe
rational -in an extended sense- choice mechanisms, inducing choice functions statisfying each
of these three axioms.
In this paper we study the order structure of the sets of choice functions satisfying each
of these axioms. Indeed, these sets are always partially ordered by the point-wise order
between functions. Moreover, we shall see that they are always lattices, with join and meet
operations which can be or not the union and intersection operations.
The interest to specify such structures is in particular linked to consensus problems. Consider
for instance individuals using choice mechanisms which induce choice functions satisfying
axiom (O). Assume that one takes as consensus the unanimity rule (x is chosen in a set if
it is chosen by all the individuals). This amounts to considering the choice function as the
intersection of the individual choice functions. But this choice function does not necessarily
satisfy axiom (O). In fact, according to a result in [1], it can be a completely arbitrary
choice function. Nevertheless, since the set of choice functions is a lattice, one can use the
unanimity rule in this lattice (the intersection operation is replaced by the meet operation)
and so to get a collective choice function having the same level of rationality as that the
individual choice functions have.
Moreover, there exist latticial theories of consensus (see e.g. [6], [11], [15], [17]) which can
be applied whenever one aggregates elements of a lattice L. In fact, there are two main
kinds of approaches and results. The first one is the classical axiomatic approach. The
consensus functions (i.e. the functions Ln 7→ L) must satisfy some ”reasonable” axioms
and the theory determines the functions satisfying such conditions. The results depend not
only on the axioms considered but also on the properties of the lattice and in particular
on the properties of dependence relations defined on the sets of its irreducible elements.
The second one is the metric approach (a generalization of Kemeny’s rule). The consensus
element is taken as a ”closest” element (relatively to a metric defined on the lattice) to the
elements to aggregate. This approach has been particularly developed in the case where the
consensus element is a ”median” element, a case where one can characterize axiomatically
the corresponding ”median procedure”.
Since our sets of choice functions are lattices one can use the results obtained on the structure of these lattices and those of the latticial theories of consensus to get results on the
aggregation problem for such choice functions. This is the process followed in this paper.
Then it should be clear that its more original part consists to determine the properties of
our lattices of choice functions, since once time these properties are known one has just to
apply the relevant general results of the latticial theories. It is also the reason explaining
why we don’t intend to give all the results which could be obtained by this approach. We
will simply give interesting results illustrating this process and we will compare them to
some previous results.
In Section 2 we begin by recalling the main notions and results of lattice theory which we
need (for some elementary facts not recalled, see e.g. [7], or [8]). In particular we shall use
2
the arrow relations defined on a lattice L and the dependence relation δ (respectively β)
defined on the set of join-irreducible (respectively meet-irreducible) elements of L.
Section 3 is devoted to studying the structure of sets of choice functions ordered by the
point-wise order between functions. We begin by noting that the set of all choice functions
on a set S is a Boolean lattice . Its atoms (respectively its coatoms) are the choice functions
cA,x (respectively cA,x ) where the choice on a set X is always empty (respectively always
equal to X) except for a set A where the choice is an element x of A (respectively A − {x}).
The lattice of choice functions satisfying axiom (H) has a nice structure since it is a distributive lattice with intersection and union as meet and join operations. We determine its meetand join-irreducible elements. Then we study the lattice of choice functions satisying axiom
(C). We show that it is atomistic (with atoms the choice functions cA,x ) and we display its
meet-irreducible elements. The study of the dependence relation δ of this lattice allows us
to prove that it is lower bounded. Since it is also atomistic, it has many other properties; for
instance, it is lower locally distributive and then lower semi-modular and ranked. Finally,
we study the much more complex lattice of choice functions satisying axiom (O). Indeed
this lattice is coatomistic (with coatoms the choice functions cA,x ) but it is not even ranked.
Moreover, its join-irreducible elements seem very difficult to characterize. Nevertheless, we
exhibit a class of such join-irreducible elements. The decomposition into ”bouquets” associated to a choice function satisying (O) is the tool used to obtain these results.
In Section 4 we begin by recalling some results of the latticial theory of consensus. Then we
apply some of these results and the results of the previous Section for the consensus of choice
functions satisfying our axioms. According to the structure of the corresponding lattice the
same type of axioms asked for the consensus functions can determine a broad class of such
functions, those associated to the families of sets called simple games (or federations) or
only a restricted class like the class of ”oligarchic” or ”co-oligarchic” consensus functions.
Concerning the metric approach, the median choice function(s) of a profile can be easy or
difficult to get according to the structure of the corresponding lattice.
In our conclusion we compare our results with those obtained by the ”russian school”. In
fact there are some identical results, some results leading to the same consensus functions
but with different axioms and some different results. We end this paper by pointing out
some generalizations of the process followed in this paper and some open problems.
2
Notations and preliminaries
General information on lattice theory may for instance be found in Davey and Priestley’s
book ([8]). Throughout this paper all lattices (or posets) are assumed to be finite. So
”lattice” means finite lattice. The following definitions, properties and notations will be
particularly useful in the sequel.
Definition 1 A poset (L, ≤) is a set L endowed with a partial order ≤.
We will often denote it only by L.
Definition 2
1. A poset L is a join semilattice if any two arbitrary elements x and
y of L have a join (i.e. a least common upper bound) denoted by x ∨ y.
2. A poset L is a meet semilattice if any two arbitrary elements x and y of L have a
meet (i.e. a greatest common lower bound) denoted by x ∧ y.
3. A lattice is a poset that is both a meet and a join semilattice.
3
A lattice can be denoted by (L, ≤, ∨, ∧),
W (L, ∨, ∧), or simply by L. Every subset A of a
(finite) lattice L has a join denoted by A and a meet denoted by ∧A. We denote by 1L or
simply by 1 (respectively by 0L or simply by 0) the greatest element (respectively the least
element) of L.
Definition 3
1. A subset S of a lattice (L, ∨, ∧) is a sub join semilattice (or sub∨-semilattice) of L if the join of two arbitrary elements of S belong to S.
2. A subset S of a lattice (L, ∨, ∧) is a sub meet semilattice (or sub-∧-semilattice)
of L if the meet of two arbitrary elements of S belong to S.
3. A subset S of a lattice (L, ∨, ∧) is a sublattice of L if S is both a sub meet semilattice
and a sub join semilattice.
Example 1 For any two elements x, y of a lattice L such that x ≤ y, the interval denoted
by [x, y] is the set [x, y] = {z ∈ L: x ≤ z ≤ y}. An interval of L is a sublattice of L.
The following well-known ([7]) result will be useful:
Lemma 1 A meet semilattice (respectively a join semilattice) having a greatest element
(respectively a least element) is a lattice.
Definition 4 The cover relation on a poset L is the binary relation defined on L by x ≺ y
if there exists no z ∈ L such that x < z < y.
If x ≺ y we say that y covers x or that x is covered by y. A poset can be vizualized by its
(Hasse) diagram: an element is represented by a point px of the plane. If y < x then py is
under px and py is linked to px if and only if y ≺ x. For instance FIG.1 shows the diagram
of a lattice L = {0, a, b, k, e, f, g, h, 1}.
W
Definition 5
1. An element j of a lattice L is join-irreducible if for A ⊆ L, j = A
implies j ∈ A, or equivalently if j covers exactly one element of L.
V
2. An element m of a lattice L is meet-irreducible if for A ⊆ L, m = A implies
m ∈ A, or equivalently if m is covered by exactly one element of L.
3. An atom of L is a (join-irreducible) element of L covering the least element 0 L .
4. A coatom of L is a (meet-irreducible) element of L covered by the greatest element
1L .
The set of all join-irreducible elements of L is denoted by JL or simply J. The set of all meetirreducible elements of L is denoted by ML or simply M . The join-irreducible (respectively
W
meet-irreducible) elements
V allow to get all the elements of L by the formula x = {j ∈ J :
j ≤ x} (respectively x = {m ∈ M : x ≤ m}).
Example 2 In the lattice L of FIG.1, JL = {a, b, f, k} and ML = {f, g, h, k}; L has two
atoms a and b, and two coatoms g and h.
4
1
g
h
e
k
a
f
b
0
FIG.1 : Lattice L = {0, a, b, e, f, g, h, k, 1}
An element x of a poset (L, ≤) is maximal if x ≤ y implies x = y.
Definition 6 Let L be a lattice and x and y be two elements of L. The arrow relations
are the binary relations defined on L by:
• x ↑ y if y is maximal in {t ∈ L: x 6≤ t}.
• x ↓ y if x is minimal in {t ∈ L: t 6≤ y}.
• x l y if x ↑ y and x ↓ y.
It is well known that x ↓ y (respectively x ↑ y) implies x ∈ JL (respectively y ∈ ML ).
Moreover JL = {j ∈ L: ∃m ∈ ML such that j ↓ m} and ML = {m ∈ L: ∃j ∈ JL such that
j ↑ m}. Henceforth we will consider that the dependence relations are defined between the
sets JL and ML : j ↑ m or j ↓ m.
Definition 7
1. The dependence relation δ on the set JL of join-irreducible elements
of a lattice L is defined by : for all j, j 0 ∈ J, jδj 0 if j = j 0 or if there exists a meetirreducible element m of L such that j ↑ m and j 0 6≤ m (see [23]).
2. The dual dependence relation β is defined on ML by: for all m, m0 ∈ M , mβm0 if
there exists j in JL satisfying j ↓ m and j 6≤ m0 .
The arrow and dependence relations of a lattice allow to give useful characterizations of
classes of lattices (see for instance Definition 9.8 below).
Definition 8 A chain of a lattice L is a subset A of L such that two arbitrary elements
of A are always comparable (i.e. x, y ∈ A implies that x ≤ y or y ≤ x). The length of a
chain of cardinality l + 1 is l. The length l(L) of a lattice L is the length of a longest chain
in L.
Example 3 In FIG.1, l(L) = 4, the length of the chain 0 ≺ a ≺ k ≺ g ≺ 1.
5
Lattices of choice functions belong to some classes of lattices that we define below.
Definition 9
1. A lattice L is distributive if it satisfies the distributive laws: for all
x, y, z ∈ L, (x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z) (or equivalently (x ∧ y) ∨ z = (x ∨ z) ∧ (y ∧ z)).
2. A lattice L is atomistic if each element x in L is a join of atoms of L (i.e. if JL is
the set of all atoms of L).
3. L is coatomistic if ML is the set of coatoms of L.
4. A lattice L is Boolean if it is distributive and atomistic.
5. A lattice L is lower semi-modular if for every x, y ∈ L, x ≺ x ∨ y implies x ∧ y ≺ y.
Dually, a lattice L is upper semi-modular if for every x, y ∈ L, x ∧ y ≺ x implies
y ≺ x ∨ y.
6. A lattice L is lower locally distributive if its length equals the number of its joinirreducible elements: l(L) = |JL | 1 .
7. A lattice L is ranked if for any x in L all maximal chains between 0 and x have
the same length (see Definition 8) denoted by r(x) and called the rank of x in L.
Equivalently L is ranked if there exists a map r from L into the set IN of integers such
that x ≺ y implies r(y) = r(x) + 1.
8. An atomistic lattice L is lower bounded if its dependence relation δ defined on J L
is acyclic (i.e if there does not exist j1 , j2 , ..., jp distinct elements of JL (p > 1) with
j1 δj2 , ...,jp−1 δjp , jp δj1 ) 2 .
The following result gives some well-known relations between the classes of lattices defined
above:
Lemma 2 A Boolean lattice is distributive, a distributive lattice is lower locally distributive,
a lower locally distributive lattice is lower semi-modular and a lower semi-modular lattice is
ranked.
An atomistic lower bounded lattice is lower locally distributive (and so lower semi-modular
and ranked).
Example 4 The lattice of FIG.1 is a distributive, not Boolean lattice.
An example of Boolean lattice is the set of all subsets of a set S (ordered by set inclusion).
This lattice has s = |S| atoms and will be denoted by 2S or 2s . Since any (finite) Boolean
lattice is isomorphic to such a lattice of subsets, a Boolean lattice with s atoms will be also
denoted by 2s .
Remark 1 Note that when L is atomistic (respectively coatomistic), j ↑ m (respectively
j ↓ m) if and only if j l m.
N.B.: The symbol + (respectively −) denotes the union of disjoint sets (respectively the
difference of sets). We will write X +x (respectively X −x) rather than X +{x} (respectively
X − {x}).
1
Such a lattice has many other equivalent definitions and is also called meet-distributive (see
e.g. [22]).
2
The notion of lower bounded lattice is defined more generally, but here we need to consider
only atomistic lower bounded lattices.
6
3
Lattices of choice functions
In this Section we deal with lattices of choice functions defined on a finite set S of cardinality
s. We will first consider the lattice of all choice functions, then successively the lattices of
all choice functions verifying each one of the axioms (H), (C) or (O). For more information
on these axioms and their significance in the theory of choice functions see for instance [1],
[3], [4], [25].
3.1
The lattice S of all choice functions
Let S be a finite set of cardinality s.
Definition 10 A choice function on S is a map c from 2S to 2S satisfying c(A) ⊆ A, for
every A ⊆ S.
So, c is a choice function if 0S ≤ c ≤ 1S where 1S is the identity function on 2S : 1S (A) = A,
for every A ⊆ S, and 0S the null function : 0S (A) = ∅, for every A ⊆ S.
We denote by S or by S(s) the set of all choice functions defined on S.
Definition 11 We define an order relation ≤ on S by: for all c, c0 ∈ S, c ≤ c0 if for every
A ⊆ S, c(A) ⊆ c0 (A).
Then S is the interval (see Example 1) [0S , 1S ] of the lattice of all functions from 2S to 2S
under the point-wise order.
Let c be a choice function on S. In order to simplify the notations, c(A) = {x} will be
denoted by c(A) = x. We introduce the following particular choice functions:
Definition 12 For every subset U ⊆ S and every x ∈ U , we set:
cU,x (X) = x if X = U and cU,x (X) = ∅ if not;
cU,x (X) = U − x if X = U and cU,x (X) = X if not.
The following result is obvious.
Proposition 1 The ordered set (S, ≤) of all choice functions on S is a Boolean lattice
(Definition 9.4) with join and meet operations defined as follow: for all c, c 0 ∈ S, for every
A ⊆ S, (c ∨ c0 )(A) = (c ∪ c0 )(A) = c(A) ∪ c0 (A) and (c ∧ c0 )(A) = (c ∩ c0 )(A) = c(A) ∩ c0 (A).
A choice function c is an atom (respectively a coatom) (Definition 5.3) of S if and only if
there exist U ⊆ S and x ∈ U such that c = cU,x (respectively c = cU,x ).
A choice function c is covered (Definition 4) by a choice function c0 if c(X) = c0 (X) except
for a set X where c(X) = c0 (X) − x (with x ∈ c0 (X)).
P
The rank (Definition 9.7) of the choice function c is r(c) =
A⊆S |c(A)| and the length
s−1
(Definition 8) of S(s) is s2 .
According to our general conventions the lattice (S, ≤, ∪, ∩) of all choice functions on S
will be denoted by S or S(s). Note that since S(s) is a Boolean lattice with s2 s−1 atoms,
s−1
(see Example 4).
|S(s)| = 2s2
7
3.2
The lattice H of choice functions verifying (H).
A choice function c on a finite set S (of cardinality s) satisfies the heritage axiom (H)
(called also heredity axiom, or Chernoff axiom, or α axiom) if:
for all A, B ⊆ S, A ⊆ B implies c(B) ∩ A ⊆ c(A).
Taking the point-wise order relation (Definition 11) on the set H of all such choice functions
we have the following result:
Proposition 2 The poset (H, ≤) of all choice functions on S satisfying (H) has the same
least and greatest element that the lattice S of all choice functions on S. It is a distributive
lattice (Definition 9.1) and so a ranked lattice (Definition 9.7 and Lemma 2).
Proof
It is obvious that the choice functions 0S and 1S satisfy (H) and it is easy to check that the
intersection and the union of two choice functions satisfying (H) verify (H). So (H, ≤) is a
sublattice of the Boolean lattice (S, ≤) and so a distributive lattice.
2
We will denote by H or H(s) the lattice of all choice functions on S satisfying (H).
In order to characterize the join-irreducible elements of the lattice H we introduce another
particular class of choice functions.
Definition 13 Let x ∈ V ⊆ S. We define the choice function c[x,V ] on S by: c[x,V ] (X) = x
if x ∈ X ⊆ V , and c[x,V ] (X) = ∅ if not.
We have the following characterization of the join-irreducible elements of the lattice H.
Proposition 3 A choice function c is a join-irreducible element (Definition 5) of the lattice
H if and only if there exist V ⊆ S and x ∈ V such that c = c[x,V ] .
Proof
First we show that c[x,V ] ∈ H. One has c[x,V ] (Y ) 6= ∅ if and only if x ∈ Y ⊆ V and
c[x,V ] (Y ) = x. Let X ⊆ Y ⊆ V . If x ∈ X then c[x,V ] (Y ) ∩ X = x = c[x,V ] (X)
and c[x,V ] (Y ) ∩ X = ∅ = c[x,V ] (X) if not. So for all X, Y ⊆ S such that X ⊆ Y ,
c[x,V ] (Y ) ∩ X = c[x,V ] (X) i.e. c[x,V ] ∈ H.
S
S
Now we show that c ∈ H implies that c =
Let V ⊆ S we
V ⊆S ( x∈c(V ) c[x,V ] ).
S
S
S
check that V ⊆S ( x∈c(V ) c[x,V ] ) ≤ c. Indeed, if X ⊆ V one has x∈c(V ) c[x,V ] (X) =
S
c(V ) ∩ X ⊆ c(X) (by (H)), and if not x∈c(V ) c[x,V ] (X) = ∅ = c(X). Conversely it is
S
S
S
obvious that c ≤ V ⊆S ( x∈c(V ) c[x,V ] ), since for every X ⊆ S, c(X) = x∈c(X) c[x,X] (X) ⊆
S
S
V ⊆S ( x∈c(V ) c[x,V ] (X)).
Since every choice function in H is union of choice functions c[x,V ] we only have to prove that
all such choice functions are join-irreducible in H, i.e. cover a unique choice function of H.
−
−
We set c−
[x,V ] (X) = x if x ∈ X ⊂ V , and c[x,V ] (X) = ∅ if not. One has c[x,V ] < c[x,V ] since
−
they are equal except in the case X = V , where c[x,V ] (V ) = c[x,V ] (V ) − x(= ∅). Moreover
−
c−
[x,V ] verifies (H). Indeed for all ∅ ⊂ X ⊆ Y ⊆ S, c[x,V ] (Y ) is not empty if and only if
8
−
x ∈ Y ⊂ V . In such a case, one has X ⊂ V . Then if x ∈ X, c−
[x,V ] (Y ) ∩ X = x = c[x,V ] (X)
−
and if x ∈
/ X, c−
[x,V ] (Y ) ∩ X = ∅ = c[x,V ] (X).
Now assume that c ∈ H and c < c[x,V ] and show that c ≤ c−
[x,V ] . c < c[x,V ] implies that
there exists A ⊆ S such that x ∈ A ⊂ V and c(A) = ∅. One has c(V ) ⊆ c[x,V ] (V ) = x. If
c(V ) = x, then x ∈ c(V ) ∩ A 6⊆ c(A) = ∅, a contradiction with (H). Then c(V ) = ∅. So
c ≤ c−
2
[x,V ] .
As consequence of this result one gets that the distributive lattice H (equivalently its set of
join-irreducible elements) has a very particular structure expressed in the following corollary.
Recall that |S| = s.
Corollary 1
1. The poset JH of join-irreducible elements of the lattice H of choice
functions verifying (H) is the union of s posets isomorphic to the Boolean lattice
2s−1 .
s−1
2. The length
and the rank (Definition 9.7) of c ∈ H is
P (Definition 8) of H is s2
r(c) = A⊆S |c(A)|. H is a covering sublattice of S: for c, c0 ∈ H, c ≺ c0 in H if and
only if c ≺ c0 in S.
3. The lattice H is isomorphic to the direct product 3 of s bounded free distributive lattices
with s − 1 generators 4 .
4. |H| = (ds−1 )s where ds−1 is the number of Sperner families
s − 1.
5
on a set of cardinality
Proof
1- For x fixed element of S there exist 2s−1 subsets V of S containing x. Then one easily
cheks that the poset {c[x,V ] , x ∈ V ⊆ S} is isomorphic to the Boolean lattice 2s−1 . Since
by Proposition 3, JH = {c[x,V ] , x ∈ V ⊆ S}, JH is the union of s posets isomorphic to the
Boolean lattice 2s−1 . In particular |JH | = s2s−1 .
2-It is well-known that the length of a distributive lattice is the number of its join-irreducible
elements. Then l(H) = s2s−1 . Since H ⊆ S is a ranked lattice having the same length that
S, its maximal chains are maximal chains of S. Then r(c) in H equals r(c) in S, i.e.
P
0
0
A⊆S |c(A)| (Proposition 1). Let c, c ∈ H. It is easy to check that c ≺ c in H if and only
0
if in S, c = c ∨ cU,x with U minimal such that x ∈ U and x ∈
/ c(U ), then if and only if c ≺ c 0
in S.
3- Since H is a distributive lattice it is isomorphic to the lattice of all ideals 6 of JH (see [7]
or [8]). On the other hand the lattice of all ideals of the union of posets is isomorphic to the
direct product of the lattices of ideals of these posets. Moreover it is well-known (see [7])
that the lattice of ideals of 2s−1 is isomorphic to the bounded free distributive lattice with
s − 1 generators F DL(s − 1)∗ . So (by 1- just above), H is isomorphic to the direct product
of s lattices, all isomorphic to the lattice of ideals of 2s−1 , and then to the direct product of
s bounded free distributive lattices with s − 1 generators.
4- It is well known (see [7]) that the cardinality of the bounded free distributive lattice with
s generators equals the number of Sperner families on S.
2
3
The direct product of s lattices (L1 , ≤1 ), ..., (Ls , ≤s ) is the poset -in fact a lattice- defined on
the set of s-tuples (x1 , ..., xs ) of L1 × ... × Ls by (x1 , ..., xs ) ≤ (x01 , ..., x0s ) if for every i = 1, ..., s,
xi ≤i x0i .
4
The bounded free distributive lattice with s generators, denoted by F DL(s)∗ , is obtained by
adding a minimum element 0 and a maximum element 1 to the free distributive lattice with s
generators (see [7] for the definition of the free distributive lattice).
5
See Definition 21 below.
6
An ideal of a poset (L, ≤) is a subset I of L such that x ∈ I and y ≤ x imply y ∈ I.
9
Example 5 Take s = 2, S = {1, 2}. One has JH(2) = {c[1,1] , c[1,12] , c[2,2] , c[2,12] } and
F DL(1)∗ = {0, g, 1} (the free distributive lattice with a unique generator g equals {g}).
These posets are represented on FIG.2.
1
g
c[1,12]
c[2,12]
c[1,1]
c[2,2]
0
The lattice F DL(1)∗ .
The poset (JH(2) , ≤).
FIG.2
One can check on FIG.3 that the lattice H(2) is isomorphic to the direct product
F DL(1)∗ × F DL(1)∗ . On this figure a choice function c is denoted by writing for each
non-empty subset A of S the subset c(A). A subset A is denoted by the sequence of its elements, and c(A) is denoted by underlining the elements of A which it contains. For instance
c = c[1,12] is written 1, 2, 12 (i.e. c(∅) = c(2) = ∅, c(1) = c(12) = 1).
1,2,12
1,2,12
1,2,12
1,2,12
1,2,12
1,2,12
1,2,12
1,2,12
1,2,12
FIG.3 : The lattice H(2) of choice functions satisfying (H).
Remark 2 Since the number of Sperner families on a set of cardinality s (i.e. |F DL(s) ∗ |)
is known up to s = 7 (see [26]), then the number of choice functions satisfying (H) is known
up to s = 8. For instance |H(3)| = 63 and |H(8)| = (2.414.682.040.998)7 .
Finally, we characterize the meet-irreducible elements of the lattice H. We need to define a
particular class of choice functions:
Definition 14 Let x ∈ V ⊆ S. We define the choice function c[x,V ] by: c[x,V ] (X) = X − x
if V ⊆ X and c[x,V ] (X) = X if not.
We have the following result:
10
Proposition 4 A choice function c is meet-irreducible (Definition 5) in H if and only if
there exist V ⊆ S and x ∈ V such that c = c[x,V ] .
Proof
1) We first show that c[x,V ] ∈ H. Let ∅ ⊂ X ⊆ Y . If V ⊆ X, then c[x,V ] (Y ) ∩ X =
(Y − x) ∩ X = X − x = c[x,V ] (X). If V 6⊆ X, c[x,V ] (Y ) ∩ X ⊆ X = c[x,V ] (X).
2) Now we prove that c = c[x,V ] is meet-irreducible, i.e. is covered by a unique element
[x,V ]
[x,V ]
of H. One define the choice function c+
on S by: c+ (X) = X − x if V ⊂ X and
[x,V ]
[x,V ]
[x,V ]
c+ (X) = X if not. One has c+
> c[x,V ] since c+
= c[x,V ] except in the case X = V ,
[x,V ]
[x,V
]
where c+ (V ) = c[x,V ] (V ) + x. Moreover c+
∈ H. Indeed let X ⊆ Y ⊆ S. One
[x,V ]
has c+ (Y ) 6= c[x,V ] (Y ) if and only if Y = V . Then one has only to consider the case
[x,V ]
[x,V ]
X ⊆ Y = V . In this case c+ (V ) ∩ X = V ∩ X = X = c+ (X), and so condition (H)
[x,V ]
is satisfied. Now, let c ∈ H with c[x,V ] < c. We have to prove that c+ (V ) ≤ c. This is
[x,V ]
equivalent to prove c+ (V ) = c(V ) = V . c[x,V ] < c implies that there exists T such that
V ⊆ T ⊆ S with c(T ) = T ⊃ c[x,V ] (T ) = T − x. If V ⊂ T and c(V ) = V − x, one has
c(T ) ∩ V = V 6⊆ c(V ) = V − x, a contradiction with (H). So we have proved that c(V ) = V .
3) In a distributive lattice the number of join-irreducible elements equals the number of
meet-irreducible elements. Since |{c[x,V ] , x ∈ V ⊆ S}| = |{c[x,V ] , x ∈ V ⊆ S}| there is no
other meet-irreducible element.
2
4
The lattice C of choice functions verifying (C)
A choice function on a finite set S (of cardinality s) verifies the concordance axiom (C)
(called also expansion axiom or γ axiom) if:
for all A, B ⊆ S, c(A) ∩ c(B) ⊆ c(A ∪ B).
We denote by C or C(s) the set of all choice functions verifying (C).
Using the following definition we give other useful characterizations of (C).
Definition 15 Let A ⊆ S, |A| ≥ 2 and
An x-covering of A is a family R = {A i ,
T x ∈ A. S
i ∈ I} of subsets of A, satisfying x ∈ i∈I Ai ⊂ i∈I Ai = A.
Example 6 The family R = {{x, y}, x ∈ A, y ∈ A − x} is an x-covering of A ⊆ S.
We have the following characterizations of (C):
Proposition 5 Let c be a choice function on S. The following properties are equivalent:
1)- c verifies (C),
T
S
2)- For all (Ai )i∈I family of subsets of S, i∈I c(Ai ) ⊆ c( i∈I Ai ),
3)- For all A ⊆ S, x ∈ A − c(A) and R x-covering of A, there exists Ai ∈ R such that
x∈
/ c(Ai ).
11
Proof
Let c be a choice function on S.
1 ⇐⇒ 2 . Indeed 1 implies 2 by recurrence on |I| and the converse is obvious.
2 ⇒ 3 . If not, there exists A ⊆ S, x ∈ AT− c(A) and anSx-covering R = (Ai )i∈I of A such
that x ∈ c(Ai ) for every Ai ∈ R. So x ∈ i∈I c(Ai ) 6⊆ c( i∈I Ai ) = c(A),
T a contradiction.
3 ⇒ 2 . Let R = (Ai )i∈I be a family ofSsubsets of S such that x ∈ i∈I c(Ai ). One has
x ∈ Ai for every i ∈ I. We set A = i∈I Ai and assume |A| ≥ 2 (if not the result is
obvious). R is an x-covering of A.SIf x ∈
/ c(A) there exists an Ai ∈ R such that x ∈
/ c(Ai )),
a contradiction. So x ∈ c(A) = c( i∈I Ai ).
2
It is easy to check that the intersection of two choice functions of C verifies (C), and it is
obvious that the trivial choice functions 0S and 1S satisfy (C). On the other hand the union
of two choice functions of C does not generally verify (C). For instance, with T
S = {1, 2, 3},
c12,1 and c13,1 (Definition 12) satisfy (C) but not c12,1 ∪c13,1 . But since C is an -semilattice
containing a greatest element, Lemma 1 gives the following result (where ≤ is the point-wise
order between choice functions).
Proposition 6 The poset (C, ≤) of all choice functions on S verifying (C)T is a lattice
with 0S (respectively 1S ) as least (respectively greatest) element. It is a sub- -semilattice
(Definition 3) of the lattice S of all choice functions on S.
Henceforth the lattice (C, ≤) is simply denoted by C or C(s).
For |S| = 2, every choice function satisfies (C). Then C(2) is the Boolean lattice S(2). This
is no longer the case for |S| ≥ 3. For instance, |C(3)| = 2744 < |S(3)| = 4096. However we
have the following result:
Proposition 7 A choice function c is an atom (Definition 5) of the lattice C if and only if
c is an atom cU,x of the lattice S. Moreover the lattice C is atomistic (Definition 9.2).
Proof
Let c = cU,x be an atom of the lattice S, i.e. a choice function such that c(U ) = x and
c(X) = ∅, for every X 6= U . It is clear that c satisfies (C). Then c is an atom of the lattice
C. Since every choice function verifying (C) contains at least an atom of the lattice S the
converse is true. Since the lattice C is contained in the atomistic lattice S and has the same
atoms it is atomistic too.
2
Since C is atomistic its join-irreducible elements (Definition 5) are its atoms determined
below. In order to determine its meet-irreducible elements we will use the characterization
of the meet-irreducible elements given in Section 2: c ∈ MC if and only if there exists a joinirreducible element j of C such that j ↑ c. Moreover, since C is atomistic, this characterization
is equivalent to write: c ∈ MC if and only if there exists an atom cU,x of C such that cU,x l c
(see Remark 1). We need the following definition, notation and lemmas:
Definition 16 For all {x, y} ⊆ V ⊆ S, we define the choice function c [xy,V ],x by
c[xy,V ],x (X) = X − x if {x, y} ⊆ X ⊆ V , and c[xy,V ],x (X) = X if not.
Remark 3 Observe that when V = {x, y}, then c[xy,V ],x = cV,x (Definition 12).
Lemma 3 For {x, y} ⊆ V ⊆ S, one has c[xy,V ],x ∈ C.
12
Proof
Let A ⊆ S such that t ∈ A−c[xy,V ],x (A), i.e t = x and c[xy,V ],x (A) = A−x. Let R = (Ai )i∈I
be an x-covering of A. Necessarily, there exist (at least one) Ai ∈ R such that {x, y} ⊆
Ai ⊂ V . So, x ∈
/ c[xy,V ],x (Ai ) and c[xy,V ],x ∈ C (by Proposition 5.3).
2
Notation:
Let V ⊆ S, x ∈ V . We set R(V, x) = {R1 , ..., Rm } the set of all x-coverings of V . When
Sk
R ∈ R(V, x), R = {V1 , ..., Vk } and for every i ∈ {1, ..., k}, x ∈ Vi and i=1 Vi = V .
Definition 17 Let V ⊆ S, x ∈ V . A family T = {T1 , ..., Tp } of subsets of V all containing
x is a transversal of the set R(V, x) of all x-covering of V if for every R ∈ R(V, x),
T ∩ R 6= ∅ (i.e., there exists Ti ∈ R).
T is a minimal transversal of R(V, x) if T is a transversal of R(V, x) and for every
Ti ∈ T , T − {Ti } is not a transversal of R(V, x) (i.e. there exists R ∈ R(V, x) such that
R ∩ (T − {Ti }) = ∅).
Lemma 4 Let V ⊆ S, x ∈ V . The family T of subsets of V all containing x is a minimal
tranversal of R(V, x) if and only if T ∈ {[xy, V ], y ∈ V − x}, i.e. if there exists y ∈ V − x
such that T = {T ⊆ S: {x, y} ⊆ T ⊆ V }.
Proof
⇐=
Let T = [xy, V ] for y ∈ V − x and R an x-covering of V . There exists at least Vi ∈ R
such that y ∈ Vi . Then {x, y} ⊆ Vi ⊆ V and Vi ∈ T . So T is a transversal of R(V, x). It
is a minimal transversal since if T 0 = T − {Vi }, {x, y} ⊆ Vi ⊆ V , the x-covering R = {Vi ,
(x + (V − Vi ))} of V is such that T 0 ∩ R = ∅ (indeed Vi ∈
/ T 0 and y ∈
/ [x + (V − Vi )] implies
0
that [x + (V − Vi )] ∈
/ T ).
=⇒
Assume that there exists a minimal transversal T of R(V, x) different from a (minimal)
transversal [xy, V ], y ∈ V − x. By minimality, for every y ∈ V − x, [xy, V ] 6⊆ T . Then for
every
y ∈ V − x there exists V (y) such that {x, y} ⊆ V (y) ⊆ V and V (y) ∈
/ T . But since
S
{V (y), y ∈ V − x} = V , R = {V (y), y ∈ V − x} is an x- covering of V . Hence R ∈ R(V, x)
and T ∩ R = ∅, a contradiction. So T = [xy, V ], for every y ∈ V − x.
2
We characterize now the relation l, which in this atomistic lattice is equal to the relation ↑
(Definition 6 and Remark 1).
Proposition 8 Let cU,x be an atom (Definition 5 and Proposition 7) of the lattice C.
If U = {x} then {c ∈ C : cx,x l c} = {cx,x }.
If |U | ≥ 2 then {c ∈ C : cU,x l c} = {c[xy,U ],x : y ∈ U − x}.
Proof
Let x ∈ U ⊆ S and c ∈ C.
If U = {x}, then cx,x l c if and only if c(x) = ∅ and c is maximal with this property. Clearly
cx,x is the unique maximal choice function such that c(x) = ∅. Then cx,x l c if and only
c = cx,x .
If |U | ≥ 2 we first claim that for every y ∈ U − x, cU,x l c[xy,U ],x . Indeed, let y ∈ U − x
and set c = c[xy,U ],x . One has x ∈
/ c(U ) and then cU,x 6≤ c. If there exists c0 ∈ C
0
0
such that x ∈
/ c (U ) and c < c , then there exists X such that {x, y} ⊆ X ⊂ U with
c(X) = X − x ⊂ c0 (X) = X. But in that case {X, (U − X) + x} is an x-covering of U with
13
x ∈ c0 (X) ∩ c0 ((U − X) + x) = c0 (X) ∩ ((U − X) + x) and x ∈
/ c0 (U ), a contradiction with
0
c ∈ C.
On the other hand, let c ∈ C with cU,x l c. We claim that c = c[xy,U ],x for y ∈ U − x.
Indeed cU,x l c implies that x ∈
/ c(U ). Hence (by Proposition 5) for every x-covering R of
U (i.e. R ∈ R(U, x)), there exists UR ∈ R such that x ∈
/ c(UR ). Let us consider the family
T of all these subsets UR , i.e. T = {UR , R ∈ R(U, x)}. T is a transversal of R(U, x).
Then T contains a minimal transversal of R(U, x), i.e., there exists y ∈ U − x such that
[xy, U ] = T (by Lemma 4). Then for every X such that {x, y} ⊆ X ⊆ U , X ∈ T and
x∈
/ c(X). Then c(X) ⊆ X − x = c[xy,U ],x (X). Hence, c ≤ c[xy,U ],x . But we have proved
above that cU,x l c[xy,U ],x . Then (by definition of l) c = c[xy,U ],x .
2
By this proposition and the fact that MC = {c ∈ C: ∃cU,x with cU,x l c}, we get the
characterization of the meet-irreductible elements of C:
Corollary 2 Let c ∈ C. The following properties are equivalent:
1- c is a meet-irreducible element (Definition 5) of C,
2- There exists x ∈ S such that c = cx,x (Definition 12) or there exist U ⊆ S, x, y ∈ U such
that c = c[xy,U ],x (Definition 16).
Remark 4 One can show that the coatoms (Definition 5) of C are exactly the choice functions c[xy,V ],x with V = {x, y} and the choice functions cx,x , that is to say all the coatoms
cV,x of the lattice S satisfying |V | ≤ 2 (see [27]).
Now, we determine the dependence relation δ (Definition 7) of the lattice C. We need the
following definitions:
Definition 18 Let 2s be the Boolean lattice (Definition 9.4) with s atoms, and 1 its greatest
element.
- The poset 2s − {1} is called the top truncated Boolean lattice and it is denoted by
(2s − 1).
- An antichain A of a poset (L, ≤) is a subset of L such that two elements of A are
incomparable.
Recall that in a lattice L, for j, j 0 ∈ JL , one has jδj 0 if and only if j = j 0 or there exists
m ∈ ML such that j ↑ m and j 0 6≤ m. We have the following results:
Proposition 9 Let ∅ ⊂ U , V ⊆ S, x ∈ U , y ∈ V , and |S| = s.
1- For cU,x 6= cV,y , cU,x δcV,y if and only if {x} = {y} ⊂ V ⊆ U .
2- The dependence relation δ (Definition 7) on the lattice C of all choice functions verifying
(C) is isomorphic to the union of s top truncated Boolean lattices (2s−1 − 1) and of an
antichain of size s.
Proof
Let ∅ ⊂ U , V ⊆ S, x ∈ U , y ∈ V .
1- Since C is atomistic cU,x δcV,y (with cU,x 6= cV,y ) if and only if there exists c ∈ MC such
that cU,x l c and cV,y 6≤ c, so if cU,x l c and y ∈
/ c(V ). By Proposition 8, cU,x l c if and
only if U = {x} and c = cx,x , or there exists t ∈ U − x such that c = c[xt,U ],x . In the case
c = cx,x , we have cV,y 6≤ cx,x if and only if y ∈
/ cx,x (V ) if and only if {x} = {y} = V . Then
cU,x = cV,y (= cx,x ) a contradiction with our assumption. Then c = c[xt,U ],x and y ∈
/ c(V ),
14
that is equivalent to {x, t} ⊆ V ⊆ U and y = x, and equivalent to {x} = {y} ⊂ V ⊆ U .
2- It is clear that the relation δ is reflexive and antisymmetric. We show that it is a transitive
relation. Let ∅ ⊂ U , V , W and x ∈ U ∩ V ∩ W such that cU,x δcV,x and cV,x δcW,x . By1-, one
has {x} ⊂ W ⊆ V ⊆ U . Then cU,x δcW,x . So δ is an order relation.
For x ∈ S and U , V ∈ [x, S], we have cU,x δcV,x if and only if {x} ⊂ V ⊆ U . Hence for every
x ∈ S, the poset {cU,x , {x} ⊂ U ⊆ S,|U | > 1} is isomorphic to the top truncated Boolean
lattice (2s−1 − 1). If x 6= y, for all U ∈ [x, S] and V ∈ [y, S], cU,x and cV,y are incomparable
according to the order δ, and then δ is isomorphic to the union of s top truncated Boolean
2
lattices (2s−1 − 1) and of the antichain formed by the s atoms cx,x , x ∈ S.
We deduce the following theorem:
Theorem 1 The lattice C of all choice functions satisfying (C) is atomistic and lower
bounded (Definition 9.8).
Proof
We have already observed in Proposition 7 that the lattice C is atomistic (Definition 9.2).
Since its dependence relation δ is an order it has no cycle. Then C is lower bounded (Definition 9.8).
2
We have the following corollary (recall that |S| = s):
Corollary 3 1- The lattice C is lower locally distributive, lower semi-modular and ranked
(Definition 9).
P
2- The length of C is s2s−1 and the rank of c ∈ C is r(c) = A⊆S |c(A)|. c ≺ c0 in the lattice
C if and only if c ≺ c0 in S.
Proof
1- This assertion results from the implications between classes of lattices recalled in Section
2 (Lemma 2).
2- It is known that the length of a lower locally distributive lattice is the number of its
join-irreducible elements. Then l(C) equals the number of atoms of C, i.e. s2 s−1 . Since
C ⊂ S is a ranked lattice having the same lengthP
that S, its maximal chains are maximal
0
0
chains of S. Then r(c) in C equals r(c) in S, i.e.
A⊆S |c(A)|. Let c, c ∈ C. c ≺ c in C if
0
0
and only if c = c ∨ cU,x i.e. if and only if c ≺ c in S.
2
4.1
The lattice O of choice functions verifying (O)
A choice function c on S (of cardinality s) verifies the outcast axiom (O) (called also
Nash’s axiom) if:
for all A, B ⊆ S, c(A) ⊆ B ⊆ A implies c(B) = c(A).
We denote by O or O(s) the set of all such choice functions.
A choice function satisfying (O) is obviously idempotent. Then, the set c(2 S ) of the images
of c is the set of the ”fixed points” of c, i.e., the family of the subsets A of S such that
c(A) = A. We denote by Ic the set of the images of c. For U ∈ Ic , we set BU = {X ⊆ S :
c(X) = U } = c−1 (U ). The set Bc = {BU : U ∈ Ic } is the canonical partition of 2S associated
15
to c. Note that this set contains always B∅ the set of subsets of X such that c(X) = ∅.
Another characterization of the choice functions verifying (O) will be useful. It needs the
following definition:
Definition 19 A bouquet on S is a family B of subsets of S which is stable by intersection
(X, Y ∈ B implies X ∩ Y ∈ B) and convexT(X ⊆ Y ⊆ Z, and X, Z ∈ B imply Y ∈ B).
The root of a bouquet B is the set U = {X, X ∈ B} of B. A bouquet with root U is
denoted by BU .
It is clear that every interval [U, V ] = {X ⊆ S: U ⊆ X ⊆ V } is a bouquet, and we have the
following obvious proposition:
Proposition 10 A family B of subsets of S is a bouquet if and only if B is the union of
intervals [U, Xi ], i = 1, ..., p.
Choice functions verifying (O) are characterized by means of the notion of bouquet:
Proposition 11 Let c be a choice function on S, Ic = {U1 , ..., Uq } the set of the images
of c and Bc = {B1 , ..., Bq } the partition associated with c: X ∈ Bi if c(X) = Ui . c satisfies
axiom (O) if and only if for all i = 1, ..., q, Bi is a bouquet with root Ui .
Proof
S
Indeed, when c satisfies (O), Bi is the bouquet BUi = k [Ui , Xk ], where for every k, Xk ∈ Bi
and is maximal (i.e. c(Xk ) = Ui and Xk ⊆ X implies c(X) 6= Ui ). The converse property
is obvious, since when c(X) = U and BU is a bouquet, U ⊆ Y ⊆ X implies that Y ∈ BU ,
hence c(Y ) = U .
2
So we obtain the following corollary:
Corollary 4 Let c be a choice function on S and X ⊆ S. If c verifies (O) and c(X) = ∅,
then for every Y ⊆ X we have c(Y ) = ∅ (i.e. c−1 (∅) is a bouquet). In particular if c(S) = ∅
then c = 0S .
It is easy to prove that the union of two choice functions of O satisfies (O) and it is obvious
that the trivial choice functions 0S and 1S verify (O). On the other hand the intersection
of two choice functions of O does not necessarily verify (O). For instance with S = {1, 2},
S
c = 1, 2, 12 and c0 = 1, 2, 12 satisfy (O) but not c ∩ c0 = 1, 2, 12. But since O is an semilattice containing a least element, Lemma 1 gives the following result (where ≤ is the
point-wise order between choice functions).
Proposition 12 The poset (O, ≤) of all choice functions on S verifying (O)
S is a lattice
with 0S (respectively 1S ) as least (respectively greatest) element. It is a sub- -semilattice
(Definition 3.1) of the lattice (S, ≤) of all choice functions on S.
Henceforth, the lattice (O, ≤) will be simply denoted by O or O(s). FIG.4 represents the
diagram of the lattice O(2) (the choice functions are written with the conventions described
16
for FIG.3). One can note that O(2) is a ranked lattice but that it is neither lower nor upper
semi-modular.
1S
1,2,12
1,2,12
1,2,12
1,2,12
1,2,12
1,2,12
1,2,12
0S
FIG.4 : The lattice O(2) of choice functions satisfying (O).
It is easy to characterize the coatoms of the lattice O.
Proposition 13 A choice function c is a coatom (Definition 5.4) of the lattice O if and
only if c is a coatom cU,x of the lattice S. Moreover the lattice O is coatomistic (Definition
9.3).
Proof
Let c = cU,x be a coatom of S, i.e. a choice function such that c(U ) = U − x and c(X) = X
for X 6= U . c belongs to O since the images of c are all the subsets X 6= U for which
BX = {X}, and the subset U − x for which BU −x = [U − x, U ]. Then c is a coatom of O,
and since every choice function satisfying (O) is contained in (at least) a coatom c U,x the
converse is true. The lattice O being contained in the coatomistic lattice S and having the
same coatoms is coatomistic too.
2
Since O is coatomistic, we have determined the meet-irreducible elements of the lattice
O. Now we search to determine the join-irreducible elements of this lattice, a not easy task
except in the case of its atoms. In order to determine these atoms we introduce the following
particular choice functions:
Definition 20 Let ∅ ⊂ U ⊆ S. We define the choice function cU on S by: cU (X) = U if
U ⊆ X ⊆ S and cU (X) = ∅ if not.
Proposition 14 A choice function c is an atom (Definition 5) of the lattice O if and only
if there exists a non-empty subset U of S such that c = cU .
Proof
First we prove
S that cU ∈ O for every ∅ ⊂ U ⊆ S. Indeed, the images of cU are ∅ and U .
Since B∅ = x∈U [∅, S − x] and BU = [U, S] are two bouquets, c ∈ O (by Proposition 11).
In order to prove that c = cU is an atom we have to show that the only choice function in
O less than c is 0S . Let c0 ∈ O such that c0 < c. For every X 6⊇ U , c0 (X) = ∅ and for every
X ⊇ U , c0 (X) ⊆ U . If c0 (S) = c(S) = U then there exists X with U ⊆ X ⊂ S such that
17
c0 (X) ⊂ U . This is impossible since c0 ∈ O and c0 (S) = U ⊆ X ⊂ S. Hence c0 (S) ⊂ U , and
since U 6⊆ c0 (S), c0 (c0 (S)) = ∅. Since c0 ∈ O, c0 is idempotent and then c0 (S) = c0 (c0 (S)) = ∅.
So c0 = 0S (by Corollary 4) and we have proved that c = cU is an atom of O.
Conversely let c be an atom of O. Then c(S) 6= ∅. We set U = c(S). For every X ⊇ U ,
c(X) = U . Then cU ≤ c. But cU and c are atoms so c = cU .
2
Now we are going to determine another class of join-irreducible elements (Definition 5) of
O. We need the following definitions and properties.
Definition 21 A family {U1 , ..., Up } of subsets of S is a Sperner family on S if Ui and
Uj are incomparable for all i 6= j, i, j ∈ {1, ..., p}.
An ordered Sperner family on S is a Sperner family {U1 , ..., Up } ordered by a strict total
order >. We denote it by (U1 > U2 > ... > Up ).
The function cU1 >U2 >...>Up associated to the ordered Sperner family (U1 > U2 > ... > Up ) is
the function on 2S defined by: cU1 >...>Up (X) = Ui if X ∈ BUi = {X ⊆ S: ∀k = 1, .., i − 1,
Uk 6⊆ X and Ui ⊆ X} and cU1 >...>Up (X) = ∅ if X ∈ B∅ = {X ⊆ S: ∀k = 1, ..., p, Uk 6⊆ X}.
In particular, when the ordered Sperner family reduces to the empty set, c∅ = 0S .
For instance (1235 > 14 > 234 > 45) is an ordered Sperner family on S = {1, 2, 3, 4, 5}.
Proposition 15 The choice function c = cU1 >...>Up associated to an ordered Sperner family
(U1 > ... > Up ) is a choice function satisfying the outcast axiom (O).
Proof
To prove this result we will successively demonstrate the following points:
1) For all i ∈ {1, ..., p}, BUi is a bouquet with root Ui .
2) The family B∅ is a bouquet with root the empty set.
3) B∅ ∪ {BUi , i = 1, ..., p} is a partition of 2S .
Let (U1 > ... > Up ) be an ordered Sperner family on S. For i ∈ {2, ..., p} we denote by
Ti the set of the minimal transversals of the family {U1 − Ui ,...,Ui−1 − Ui } (note that by
definition Uk − Ui 6= ∅ for every k < i).
S
1) By definition BU1 = [U1 , S] and we claim that BUi = T ∈Ti [Ui , S −T ] for every i = 2, ..., p.
Let i > 1 and X ∈ BUi . Then Ui ⊆ X and Uk 6⊆ X for every k = 1, ..., i − 1. Hence
there exists xk ∈ Uk − Ui such that X ⊆ S − xk . Then X ⊆ S − {x1 , ..., xi−1 } with
{x1 , ..., xi−1 } transversal of {U1 − Ui , ..., Ui−1 − Ui }. Let T ∈ Ti be a minimal transversal
of {U1 − Ui , ..., Ui−1 − Ui } such that T ⊆ {x1 , ..., xi−1 }. Then X ⊆ S − T . So X ∈
S
T ∈Ti [Ui , S − T ]. S
Conversely if X ∈ T ∈Ti [Ui , S − T ], then there exists T ∈ Ti such that Ui ⊆ X ⊆ S − T .
Moreover one has Uk 6⊆ X for every k = 1, ..., i − 1. If not Uk ⊆ X ⊆ S − T , and then
Uk ∩ T = (Uk − Ui ) ∩ T = ∅ a contradiction with T ∈ Ti . So X ∈ BUi .
2) If X ∈ B∅ , ∅ ⊆ Y ⊆ X obviously implies Y ∈ B∅ . Then B∅ is the bouquet equal to the
union of intervals [∅, Xi ] where Xi is maximal in B∅ .
3) Let BUi the bouquet with root Ui . For all i 6= j, BUi ∩ BUj = ∅. Indeed, if for example
i < j, X ∈ BUi implies that Ui ⊆ X, and X ∈ BUj implies that Ui 6⊆ X. On the other hand
for every X ∈ B∅ one has Uk 6⊆ X for every k, then B∅ ∩ BUk = ∅ for every k. And since it
is clear that for every X ∈ 2S , X ∈ BUi for an i ∈ {1, ..., p} or X ∈ B∅ , one has the result.
Then the canonical partition Bc = {BU1 , ..., BUp , B∅ } of 2S associated to c is formed of
bouquets and by Proposition 11, c ∈ O.
2
Henceforth we will simply denote Bc = {B1 , ..., Bp , B∅ }. We will also need to consider
Bc∗ = {B1 , ..., Bp }.
18
Example 7 c1235>14>234>45 is the choice function associated to the ordered Sperner family
1235 > 14 > 234 > 45. It is represented by its set Bc∗ = {B1 = B1235 , B2 = B14 , B3 = B234
and B4 = B45 } of bouquets on FIG.5.
We are going to show that some choice functions cU1 >...>Up are join-irreducible elements of
(O). In fact in Theorem 2 below, we will characterize such choice functions. But we need
first to give some definitions and to prove several technical results. In particular we need
to introduce a graph defined on the set Bc∗ = {BU1 , ..., BUp } of bouquets of c, different from
B∅ , and to consider saturated sets and sources of this graph.
Definition 22 Let c = cU1 >...>Up be a choice function associated to an ordered Sperner
family (U1 > ... > Up ).
1. We define a digraph Gc = (Bc∗ , ∆) by: Bc∗ = {B1 , ..., Bj , ..., Bp } is the set of the
bouquets of c which are different from B∅ (Bj = c−1 (Uj ) = BUj ); the relation ∆ is
defined by Bk ∆Bj if k > j and there exists X ∈ Bj with X ⊃ Uk .
2. A subset B of Bc∗ is a saturated set of the digraph Gc = (Bc∗ , ∆) if Bk ∈ B and
Bk ∆Bj imply Bj ∈ B.
3. A source of the digraph Gc = (Bc∗ , ∆) is a bouquet Bi ∈ Bc∗ such that there does not
exist Bj ∈ Bc∗ (i < j) with Bj ∆Bi .
4. We set I = {1, ..., p}. Let ∅ ⊆ J ⊆ I. We define the choice function cJ = cJU1 >...>Up
on S by: cJU1 >...>Up (X) = Uj if j ∈ J and X ∈ Bj , and cJU1 >...>Up (X) = ∅ if not.
In other words cJ is identical to c on all the bouquets Bj where j ∈ J and the choice
is empty if not. In particular, c∅ = 0S and cI = c.
Example 8 The digraph Gc = (Bc∗ , ∆) for the choice function c = c1235>14>234>45 is represented on FIG.5. B4 is a source of Gc and {B3 , B2 , B1 } is a saturated set of Gc .
Remark 5 Observe that the digraph Gc is contained in the graph of the linear order Bp >
... > B2 > B1 . So Bp is a source of Gc and satisfies Bk ∆B1 , ∀k = 2, ..., p (since S ∈ B1 ).
Note also that one can give an equivalent definition of the relation ∆:
Bk ∆Bj if and only if j = 1, or k > j and ∀i = 1, ..., j − 1, Ui 6⊆ Uj ∪ Uk .
Lemma 5 Let c = cU1 >...>Up be a choice function in O, and c0 a choice function. The two
following properties are equivalent:
1. c0 < c and c0 ∈ O,
2. There exists J ⊂ I = {1, ..., p} such that c0 = cJU1 >...>Up and B J = {Bj , j ∈ J} is a
saturated set of the digraph Gc .
In particular c0 ≺ c and c0 ∈ O if and only if c0 = cI−j and B I−j is a saturated set of the
digraph Gc .
Proof
1. =⇒ 2.
a)- Assume that c0 < c and c0 not identical to c on a bouquet Bj . We are going to show
that for every X ∈ Bj , c0 (X) = ∅. By definition of the choice function c there exists
X ∈ Bj such that c0 (X) ⊂ Uj . c(c0 (X)) = Ui (with i 6= j) is impossible since it would
imply Ui ⊆ c0 (X) ⊂ Uj , a contradiction with the hypothesis that (U1 , ..., Up ) is a Sperner
19
family. Then c(c0 (X)) = ∅ and c0 (X) = c0 (c0 (X)) ⊆ c(c0 (X)) = ∅ imply c0 (X) = ∅. Since
c0 (X) = ∅ ⊂ Uj ⊆ X and c0 ∈ O, one has as well c0 (Uj ) = ∅. Let now Y ∈ Bj with Y 6= X.
c0 (Y ) = Uj is impossible since then c0 (Uj ) = c0 (c0 (Y )) would be equal to Uj 6= ∅. Then
c0 (Y ) ⊂ Uj and like above c0 (Y ) = ∅.
b)- Assume that B J is not saturated. There exists Bj ∈ B J , Bi ∈ Bc with Bj ∆Bi and
Bi ∈
/ B J . Then one has cJ (X) = ∅ for every X ∈ Bi . But Bj ∆Bi implies the existence
of X ∈ Bi such that X ⊃ Uj . Hence ∅ = cJ (X) ⊂ Uj ⊂ X, but since cJ (Uj ) = Uj , one
contradicts c ∈ O.
2. =⇒ 1.
Let c0 = cJU1 >...>Up be a choice function associated with B J a saturated subset of the digraph
Gc . It is obvious that c0 < c. We claim that c0 (X) ⊆ Y ⊆ X implies c0 (X) = c0 (Y ), i.e.
c0 ∈ O.
If X ∈ Bj ∈ B J (or X ∈ B∅ ), c0 (X) = c(X) = Uj (or c0 (X) = ∅). Bj convex implies Y ∈ Bj
and then c0 (Y ) = Uj (or ∅).
If X ∈ Bi and Bi ∈
/ B J , c0 (X) = ∅. One has two cases:
a)- Y ∈ Bk with Bk ∈
/ B J . Then c0 (Y ) = ∅ by the definition of c0 = cJU1 >...>Up .
b)- Y ∈ Bj with Bj ∈ B J . Then j < i is impossible since Uj ⊆ Y ⊂ X would imply a
contradiction with X ∈ Bi . If j > i, then there exists X ∈ Bi with X ⊃ Uj , i.e. Bj ∆Bi . But
since B J is saturated one has Bi ∈ B J , a contradiction. So this second case is impossible. 2
Now we can give a characterization of some join-irreducible elements of the lattice O. By
remark 5, Bp is always a source of the digraph Gc . The following result shows that the
cU1 >...>Up join-irreducible elements of O are characterized by the fact that Bp is the unique
source of Gc .
Theorem 2 Let (U1 > .... > Up ) be an ordered Sperner family on S and Gc =
({B1 , ..., Bp }, ∆) the digraph associated to the choice function c = cU1 >...>Up . The two
following properties are equivalent:
1. c = cU1 >...>Up ∈ JO ,
2. the digraph Gc has Bp as unique source.
Moreover in this case the choice functions 0S , cU1 , cU1 >U2 ,...,cU1 >...>Up form a covering
chain of join-irreducible elements (Definition 5) of the lattice O: 0 S ≺ cU1 ≺ cU1 >U2 ≺ ... ≺
cU1 >...>Up .
In particular, c = cU1 >...>Up ∈ JO if the sets U1 , ..., Up are pairwise disjoint, and for p = 2,
cU1 >U2 is always a join-irreducible element covering cU1 .
Proof
By Lemma 5, c0 ≺ c in O if and only if c0 = cI−j (I = {1, ..., p}) with B I−j saturated set of
the digraph Gc . Now it is obvious that B I−j is saturated if and only if Bj is a source of Gc .
Since Bp is a source of Gc , B I−p is saturated and c0 = cU1 >...>Up−1 is covered by c. Then c
is join-irreducible if and only if c0 is the unique choice function covered by c and if and only
if Bp is the unique source of Gc .
Now take c0 = cU1 >...>Up−1 . Since the digraph Gc0 is the restriction of the digraph Gc to
{B1 , ..., Bp−1 }, it has also a unique source namely Bp−1 . Then iterating the above reasoning
one obtains that cU1 >...>Up−2 is join-irreducible and finally the stated result. If Ui ∩ Uj = ∅
for all i 6= j, it is easy to check that Bp is the unique source of Gc . If p = 2, it is obvious
that B2 is the unique source of Gc .
2
20
Example 9 The digraph Bc associated to the choice function c = c1235>14>234>45 has a
unique source (FIG.5). By Theorem 2, there exists a covering chain of join-irreducible
elements of the lattice O represented on this same Figure.
Remark 6 The set JO of join-irreducible elements of the lattice O contains strictly the set
of join-irreducible cU1 >...>Up characterized in Theorem 2. Indeed let us consider the choice
function defined on S = {1, 2, 3} by c(123) = 123, c(12) = c(1) = 1, c(23) = c(2) = 2 and
c(13) = c(3) = 3. It is easy to see that c is join-irreducible in O, but c cannot be obtained as
cU1 >...>Up . Moreover in the lattice O(3) one can find a maximal chain of length (Definition
8) 5 from 0S to 1S passing by c and another maximal chain from 0S to 1S of length 7. Then
for s ≥ 3, O(s) is not a ranked lattice (Definition 9). One can show that the maximum
length of a maximal chain in O(s) is 2s − 1.
The following theorem summarizes the properties of the lattice O (recall that |S| = s):
Theorem 3 The lattice O of all choice functions satisfying axiom (O) is coatomistic (Definition 9.3) (with the coatoms cU,x of the lattice S as coatoms). Its join-irreducible elements contain all the choice functions c = cU1 >...>Up such that the associated digraph
Gc = ({B1 , ..., Bp }, ∆) has Bp as unique source. The length of O is 2s − 1. For s = 2 the lattice O(2) is ranked (Definition 9.7) but neither upper semi-modular nor lower semi-modular
(Definition 9.5) . For s ≥ 3 the lattice O(s) is not ranked.
21
12345
1235
1245 1234 1345
2345
B1
124
145
134
234
245
345
B3
14
45
B2
B4
c1235>14>234>45
c1235>14>234
B1
B2
6I 6
B4
c1235>14
B3
c1235
Gc = (Bc , ∆)
0S
FIG.5: The choice function associated to (1235 > 14 > 234 > 45),
and the associated covering chain of join-irreducible elements of O.
5
The consensus of choice functions
As said in the introduction we are going now to use the previous results on lattices of choice
functions and the general results of the latticial consensus theory to derive results on the
consensus of choice functions. We will consider successively the axiomatic approach (Section 5.1) then the metric approach (Section 5.2). In each case we will begin by giving some
results of the latticial theory then we will apply them to the consensus of choice functions.
We shall use the following definitions and notations:
Let L be a finite lattice.
A n-profile or simply profile of L is a n-tuple Π = (x1 , ..., xi , ..., xn ) of elements of L. We
denote by Ln the set of all n-profiles of L.
The consensus problem on L consists of summarizing arbitrary profiles Π of L by one or
several elements of L.
A consensus function F on L is a map from Ln into L or 2L , that is for every Π in Ln ,
F (Π) belongs to L or is a subset of L. The problem is to find ”good” consensus functions.
22
5.1
5.1.1
The axiomatic approach
Axiomatic latticial consensus theory
First let us define some classes of latticial consensus functions F : Ln 7→ L (L lattice). A
federation (called also simple game) on N = {1, ..., i, ..., n} is a family F of subsets of N
satisfying [G ∈ F, K ⊇ G] imply [K ∈ F].
Definition 23
1. A federation consensus function is aWconsensus
V function F on L
such that there exists a federation F for which F (Π) = K∈F ( i∈K xi ) for every Π
in Ln .
2. A meet-projection consensus function is a consensusVfunction for which there exists
K ⊆ N (K 6= ∅) such that for each Π in Ln , F (Π) = i∈K xi .
V
In particular, F (Π) = xi is a projection when |K| = 1, and F (Π) = i∈N xi is the
Pareto (or unanimity) consensus function when K = N .
Note that the meet-projection consensus functions are federation consensus functions (take
F = {G ⊆ N : K ⊆ G}). In social choice theory the meet-projection (respectively projection) functions are often called oligarchic (respectively dictatorial 7 ).
Using the dual operations on the lattice L one defines dual consensus functions:
Definition 24
1. A co-federation consensus function is aVconsensus
W function F on
L such that there exists a federation F for which F (Π) = K∈F ( i∈K xi ) for every
Π in Ln .
2. A join-projection consensus function is a consensusWfunction for which there exists
K ⊆ N (K 6= ∅) such that for each Π in Ln , F (Π) = i∈K xi .
W
In particular F (Π) = xi is a projection when |K| = 1, and F (Π) = i∈N xi is the
co-Pareto consensus function when K = N .
Note that join-projection consensus functions are co-federation consensus functions (take
F = {G ⊆ N : K ⊆ G}). These functions can be also called co-oligarchic.
Next we give some Arrovian axioms of the latticial consensus theory.
Let j be a join-irreducible element of the lattice L. We denote by Nj (Π) the set Nj (Π) =
{i ∈ N : j ≤ xi }.
Definition 25 A consensus function F on the lattice L is:
1. JL -neutral monotonic if for all j, j 0 in JL and Π, Π0 in Ln , one has: [Nj (Π) ⊆
Nj 0 (Π0 )] =⇒ [j ≤ F (Π) =⇒ j 0 ≤ F (Π0 )].
2. JL -decisive if for every j in JL and for all Π, Π0 in Ln , [Nj (Π) = Nj (Π0 )] =⇒ [j ≤
F (Π) ⇐⇒ j ≤ F (Π0 )].
7
In fact in social choice theory one has to consider preference agregation functions that are
absolute dictatorial (i.e. projections) and preference agregation functions that are dictatorial. But
we will not have to make this distinction here.
23
3. Paretian if for each Π in Ln ,
V
i∈N
xi ≤ F (Π).
We define below dual axioms bearing on the meet-irreducible elements on L. For m in M L ,
Nm (Π) = {i ∈ N : m ≥ xi } .
Definition 26 A consensus function F on the lattice L is:
1. ML -neutral monotonic if for all m, m0 in ML and Π, Π0 in Ln , one has: [Nm (Π) ⊆
Nm0 (Π0 )] =⇒ [m ≥ F (Π) =⇒ m0 ≥ F (Π0 )].
2. ML -decisive if for every m in ML and for all Π, Π0 in Ln , [Nm (Π) = Nm (Π0 )] =⇒
[m ≥ F (Π) ⇐⇒ m ≥ F (Π0 )].
W
3. co-Paretian if for each Π in Ln , i∈N xi ≥ F (Π).
The consensus functions defined on a lattice L depend on the chosen axioms and on the
structural properties of the considered lattice, and in particular on the properties of the
dependence relations δ (defined on the set JL of all join-irreducible elements of L) and β
(defined on the set ML of its meet-irreducible elements). The following theorem give some
general results of the axiomatic latticial consensus theory (see [15], [21]).
Theorem 4 Let L be a lattice and F be a consensus function on L.
1. If L is distributive (Definition 9.1), then F is JL -neutral monotonic if and only if F
is a federation consensus function.
2. If L is not distributive, then F is JL -neutral monotonic and Paretian if and only if F
is a meet-projection.
3. If the dependence relation δ on JL (Definition 7.1) is strongly connected, then F is
JL -decisive and Paretian if and only if F is a meet-projection consensus function.
Note that the condition on the dependence relation δ in point 3 of the above Theorem implies
that L is not distributive (indeed, a lattice L is distributive if and only if the dependence
relation δ on JL is the order relation on JL (cf. [23]).
A dual theorem exists using dual axioms and the dual dependence relation β defined on the
set ML of all meet-irreducible elements of L. In particular one obtains a characterization of
the co-federation and join-projection consensus functions.
Theorem 5 Let L be a lattice and F be a consensus function on L.
1. If L is distributive (Definition 9.1), then F is ML -neutral monotonic if and only if F
is a co-federation consensus function.
2. If L is not distributive, then F is ML -neutral monotonic and co-Paretian if and only
if F is a join-projection.
3. If the dependence relation β on ML (Definition 7.2) is strongly connected, then F is
ML -decisive and co-Paretian if and only if F is a join-projection consensus function.
Remark 7 If a lattice L is distributive any co-federation consensus function is equal to
a federation consensus function. So in this case Theorem 4 and 5 (points 1) give two
characterizations of the same class of consensus functions.
We now deal with the consensus functions of choice functions verifying the axioms (H), (C)
or (O). Then Π = (c1 , ..., ci , ..., cn ) is a profile of choice functions all belonging either to H,
or to C or to O. A consensus function is now -for instance in the case of H- a function F
from Hn to H. Then F (Π) is denoted by c (and F (Π0 ) by c0 ).
24
5.1.2
The consensus of choice functions satisfying the heredity property
(H)
Recall (Proposition 2) that the poset (H, ≤) of choice functions satisfying (H) is a distributive
lattice (Definition 9.1). Since (Proposition 3) JH = {c[x,A] , x ∈ A ⊆ S} and (Proposition 4)
MH = {c[x,A] , x ∈ A ⊆ S}, it is straightforward to check that the axioms given in Definition
25 and Definition 26 become the following ones:
A consensus function F on H is:
• JH -neutral monotonic if for all Π, Π0 in Hn , for all subsets A, B of S, for every x
in A and for every y in B, [{i ∈ N : ∀X with x ∈ X ⊆ A, x ∈ ci (X)} ⊆ {i ∈ N : ∀Y
with y ∈ Y ⊆ B, y ∈ c0i (Y )}] implies [∀X with x ∈ X ⊆ A, x ∈ c(X)] =⇒ [∀Y with
y ∈ Y ⊆ B, y ∈ c0 (Y )].
• MH -neutral monotonic if for all Π, Π0 in Hn for all subsets A, B of S, for every x
in A and for every y in B, [{i ∈ N : ∀X with A ⊆ X, x ∈
/ ci (X)} ⊆ {i ∈ N : ∀Y
/ c(X)] =⇒ [∀Y with B ⊆ Y ,
with B ⊆ Y , y ∈
/ c0i (Y )}] implies [∀X with A ⊆ X, x ∈
y∈
/ c0 (Y )].
• Paretian if for every Π in Hn , for every subset A of S and for every x in A, [∀i ∈ N ,
x ∈ ci (A)] =⇒ [x ∈ c(A)].
• co-Paretian if for every Π in Hn , for every subset A of S and for every x in A,
[∀i ∈ N , x ∈
/ ci (A)] =⇒ [x ∈
/ c(A)].
By Theorem 4.1, Theorem 5.1 and Remark 7 the following result holds:
Theorem 6 A consensus function F on H is JH -neutral monotonic and Paretian if and
only if F is MH -neutral monotonic and co-Paretian and if and only if F is a federation
n
consensus function (i.e. if there
S exists
T a federation F on N such that for every Π in H
and for every A ⊆ S, c(A) = K∈F [ i∈K ci (A)]).
Remark 8 In the case of the Boolean lattice (Definition 9.4) S of all choice functions on S
one could write the same result but with axioms of neutrality monotonicity using the atoms
and the coatoms of S, i.e. the choice functions cA,x and cA,x . For instance, in the case of
JS , this axiom is the same that the axiom of JC -neutrality monotonicity given just below.
5.1.3
Consensus of choice functions satisfying the concordance axiom
(C)
Recall (Theorem 1) that the poset (C, ≤) of choice functions satisfying (C) on S is a lower
bounded lattice (Definition 9.8) (not distributive) and that its dependence relation δ is not
strongly connected (Proposition 9). Since (Proposition 7) JC = JS the axiom of J-neutrality
monotonicity (Definition 25) becomes:
A consensus choice function c on C is JC -neutral monotonic if for all Π, Π0 in C n , for all
subsets A, B of S, for every x in A and for every y in B one has:
[{i ∈ N : x ∈ ci (A)} ⊆ {i ∈ N : y ∈ c0i (B)}] =⇒ [x ∈ c(A) =⇒ y ∈ c0 (B)].
Then Theorem 4 induces the following result:
25
Theorem 7 A consensus function F on C is JC -neutral monotonic and Paretian if and
only if FT is oligarchic (there exists K ⊆ N such that for every Π ∈ C and for every A ⊆ S,
c(A) = i∈K ci (A)).
Taking the corresponding axioms for MC one would obtain a dual result, i.e. a characterization of the co-oligarchic consensus functions on C.
5.1.4
Consensus of choice functions satisfying the outcast axiom (O)
Recall (Proposition 13) that MO = MS = {cA,x : x ∈ A ⊆ S}, with cA,x (X) = A − x if
X = A and cA,x (X) = X if not. Definition 26 induces the following axioms:
A consensus choice function c on O is:
• MO -neutral monotonic if for all Π, Π0 in On , for all subsets A, B of S, for every
x in A and for every y in B, [{i ∈ N : x ∈
/ ci (A)} ⊆ {i ∈ N : y ∈
/ c0i (B)}] implies
0
[x ∈
/ c(A)] =⇒ [y ∈
/ c (B)].
• co-Paretian if for every Π in O n , for every A in S and for every x in A, [∀i ∈ N ,
x∈
/ ci (A)] =⇒ [x ∈
/ c(A)].
By Theorem 5 the following result holds:
Theorem 8 A consensus function F on O is MO -neutral monotonic and co-Paretian if
and only if F is co-oligarchic
(i.e. if there exists K ⊆ N such that for every Π ∈ O n and
S
for every A ⊆ S, c(A) = i∈K ci (A)).
We conjecture that the dependence relation β on MO is strongly connected. If this conjecture
was true, one could replace in Theorem 8 the MO -neutrality monotonicity axiom by the
simpler MO -decisivity axiom.
5.2
5.2.1
The metric approach
Metric latticial consensus theory
In this approach we take as consensus of a profile Π = (x1 , ..., xn ) of elements of a lattice
L an element x minimizing
Pn the ”remoteness” of x to the elements of Π. If the remoteness
is computed as D(x) = i=1 d(x, xi ), where d is a metric defined on L, such an element is
called a d-median of Π. Here we take as metric d on L the minimum path length metric in
the (unoriented) covering graph G(L) of L: x and y are linked by an edge in G(L) if and only
if x ≺ y or y ≺ x. d(x, y) is the minimum number of edges of a path of G(L) between x and
y. Then a d-median of Π will be simply called a median of Π. Such a median always exists
(since L is finite) but it is not necessarily unique. We shall denote by M edL (Π) or simply
M ed(Π) the set of medians of Π and we call median rule the consensus rule associating to
each profile its set of medians (for the median rule in general, see e.g. [5]).
These notions are illustrated on FIG.1. This figure was already used to show the diagram
of a lattice L. If one considers now that a link between two elements represents an edge
and not a (directed) arc, this same Figure represents the covering graph G(L) of L. On this
figure the distance d(k, b) is 3 (obtained for three different paths). The profile Π = (b, g, h, k)
has two medians e and g with D(e) = D(g) = 5.
Note that in arbitrary lattices neither the computation of the minimum path length metric
26
nor a fortiori the computation of the medians are necessarily easy. But in the case of
lower semi-modular lattices (Definition 9.5) the minimum path length metric is given by the
following formula: d(x, y) = 2r(x∨y)−r(x)−r(y) where r is the rank function (Definition 9.7)
on L 8 . In particular for distributive lattices this formula becomes d(x, y) = r(x∨y)−r(x∧y).
For instance on FIG.1 which represents a distributive lattice, d(k, b) = r(g)−r(0) = 3−0 = 3.
Theorem 9 below shows that the medians are easy to compute in a distributive lattice. We
use the following notations:
For all Π ∈ Ln and j ∈ JL (respectively m ∈ ML ) we denote by nj (Π) (respectively nm (Π))
the number of elements of Nj (Π) such that nj (Π) = |Nj (Π)| (respectively Nm (Π) such that
nm (Π) = |Nm (Π)|).
Definition 27 Let Π ∈ Ln . We
W V
• cΠ = { i∈K xi : K ⊆ N ,
W V
• c∗Π = { i∈K xi : K ⊆ N ,
V W
• c0Π = { i∈K xi : K ⊆ N ,
define by two equivalent ways the following elements:
W
|K| > n/2} = {j ∈ JL : nj (Π) > n/2}.
W
|K| ≥ n/2} = {j ∈ JL : nj (Π) ≥ n/2}.
V
|K| > n/2} = {m ∈ ML : nm (Π) > n/2}.
cΠ and c∗Π define consensus functions on Ln called the strict majority rule and the majority
rule respectively. Indeed they generalize the usual majority rules defined for profiles Π =
(R1 , ..., Rn ) of
S binary relations defined on a set E.S For
T instance the strict majority rule is
RM AJ (Π) = {(x, y) ∈ E 2 : |NΠ (x, y)| > n/2} = { i∈K Ri , K ⊆ N , |K| > n/2}.
Note that one always has cΠ ≤ c∗Π ≤ c0Π , and so one can consider the two nested intervals
[cΠ , c∗Π ] and [cΠ , c0Π ].
We have the following results (cf [11]):
Theorem 9 Let L be a lattice.
1. L is distributive (Definition 9.1) if and only if for every Π ∈ Ln , M ed(Π) = [cΠ , c0Π ] =
[cΠ , c∗Π ].
2. L is lower semi-modular (Definition 9.5) if and only if for every Π ∈ Ln and x ∈
M ed(Π), x ≤ c0Π .
So in a distributive lattice the medians of a profile are exactly the elements in the interval
determined by the two majority rules. Note that if n is odd these two elements coincide
and so there is a unique median. We shall show in Remark 10 below how to compute the
medians in a distributive lattice.
Remark 9 In a lower semi-modular lattice one can give a lower bound for the medians of
a profile (see [16]).
In the case of a distributive lattice the median rule, which associates to each profile its set
of medians can be characterized.SWe need some definitions:
First we denote by L∗ the set n>0 Ln of all profiles of n elements of L, where n is an
integer greater than 0. A consensus rule is a map F : L∗ 7→ 2L − {∅} associating to each
profile a non-empty subset of L. If Π = (x1 , ..., xn ) and Π0 = (y1 , ..., yp ) are two profiles,
ΠΠ0 denotes the concatened profile (x1 , ...xn , y1 , ..., yp ).
A consensus rule F is:
8
In Lemma 2 it has been recalled that a lower semi-modular lattice is ranked.
27
• consistent if F (Π) ∩ F (Π0 ) 6= ∅ implies that F (ΠΠ0 ) = F (Π) ∩ F (Π0 ) (for all Π,
Π0 ∈ L∗ ).
• unanimous if F (x, ..., x) = x, for every x ∈ L.
• quasi-Condorcet if for every Π = (x1 , ..., xn ) ∈ L∗ , for every j ∈ JL such that
nj (Π) = n/2, [x ∨ j ∈ F (Π) if and only if x ∨ j − ∈ F (Π)] (where j − is the unique
element covered by j).
We have the following result ([6], [18]), ([18] correcting an error in the statement of the
quasi-Condorcet condition contained in [6]).
Theorem 10 Let L be a distributive lattice (Definition 9.1). A consensus rule L ∗ 7→ 2L −
{∅} is the median rule if and only if it is consistent, unanimous and quasi-Condorcet.
Remark 10 In order to compute the medians of a profile Π ∈ Ln when L is a distributive
lattice one can use theWfollowing procedure (see [11]). First one computes the least median c Π
by the formula cΠ = {j ∈ JL : nj (Π) > n/2}. Then one computes the set E(Π) = {j ∈WJL :
nj (Π) = n/2} (an empty set if n is odd). The medians of Π are all the elements c Π ∨ ( G)
where G is any subset of E(Π).
Example 10 We illustrate the computation of medians described in the above Remark in
the case of the distributive lattice L of FIG.1. We take Π = (b, g, h, k). One has
n=4
JL = {a, b, f, k}
na = W
|{g, h, k}| = 3, nb = |{b, g}| = 3, nk = |{g, k}| = 2, nf = |{h}| = 1
cΠ = {j ∈ JL : nj (Π) > 2} = a ∨ b = e
E(Π) = {j ∈ JL : nj (Π) = 2} = {k}.
W
Since |E(Π)| = 1, M ed(Π) = {cΠ , V
cΠ ∨ k} = [e, g]. And one cheks that c∗Π = {j ∈ JL :
nj (Π) ≥ 2} = a ∨ b ∨ k = g = c0Π = {m ∈ ML : nm (Π) > 2}.
We shall illustrate how to apply the results of this section in the case of H and C.
5.2.2
The median rule for the choice functions satisfying the heredity
axiom (H)
First we precise the minimum path lenght metric on H. Since H is a covering sublattice of S
(Corollary 1) the distance between two choice functions in H is the same that their distance
0
0
0
in
P S. Then 0since S is a 0Boolean lattice (Definition 9.4), d(c, c ) = r(c ∪ c ) − r(c ∩ c ) =
A⊆S [|c ∪ c (A)| − |c ∩ c (A)|].
We use the same notations that in 5.2.1 except that for c[x,A] in JH , we write n[x,A] (Π)
(instead of nc[x,A] ). So for Π ∈ Hn and c[x,A] ∈ JH :
• n[x,A] (Π) = |{i ∈ N : c[x,A] ≤ ci }| = |{i ∈ N : x ∈ X ⊆ A implies x ∈ ci (X)}|
S T
S
• cΠ = { i∈K ci : K ⊆ N , |K| > n/2} = {c[x,A] ∈ JH : n[x,A] > n/2}
S
S T
• c∗Π = { i∈K ci : K ⊆ N , |K| ≥ n/2} = {c[x,A] ∈ JH : n[x,A] ≥ n/2}
28
n
The set of medians of a profile
S Π ∈n H is denoted by M edH (Π).
∗
We denote by H the set n>0 H and consider the following axioms for a consensus rule
F : H∗ 7→ 2H − {∅}:
• F is consistent if F (Π) ∩ F (Π0 ) 6= ∅ implies that F (ΠΠ0 ) = F (Π) ∩ F (Π0 ) (for all Π,
Π0 ∈ H∗ ).
• F is unanimous if F (c, ..., c) = c, for every c ∈ H.
• F is quasi-Condorcet if for every Π ∈ H∗ , for every c[x,A] ∈ JH such that n[x,A] (Π) =
n/2, [c ∪ c[x,A] ∈ F (Π) if and only if c ∪ c−
[x,A] ∈ F (Π)].
Then applying Theorems 9 and 10, and noting by [cΠ , c∗Π ]H the interval determined by cΠ
and c∗Π in the lattice H, one gets:
Theorem 11 Let H be the lattice of all choice functions verifying the heredity axiom.
• For every Π ∈ Hn , M edH (Π) = [cΠ , c∗Π ]H .
• A consensus rule H∗ 7→ 2H −{∅} is the median rule associating to each profile Π the set
of medians M edH (Π) if and only if it is consistent, unanimous and quasi-Condorcet.
Example 11 Consider the lattice H(2) represented in FIG.3 and take Π = (1, 2, 12; 1, 2, 12;
1, 2, 12; 1, 2, 12). In order to compute the medians of this profile one can remark that the
lattice H(2) is isomorphic to the lattice of FIG.1, and that in this isomorphism the above
profile Π corresponds to the profile Π = (b, g, h, k) considered in Example 10. Since the
medians of this last profile are e and g the medians of the first profile are 1, 2, 12 and 1, 2, 12.
Note that in this example there exists no other choice functions between the two medians
of Π. More generally, since S is a Boolean lattice, the interval [cΠ , c∗Π ] is the set of medians
of Π in S. When Π ∈ Hn , the set [cΠ , c∗Π ]H of medians of Π in H is [cΠ , c∗Π ] ∩ H, which is
generally a strict subset of [cΠ , c∗Π ].
5.2.3
The median rule for choice functions satisfying the concordance
axiom (C)
Recall (Corollary 3) that the lattice C of choice functions on S satisfying the concordance
axiom (C) is lower semi-modular
(Definition 9.5). We denote by M edC (Π) the set of medians
T
of a profile Π ∈ C n . c0Π = {c ∈ MC : nc (Π) > n/2} where (Corollary 2) MC = {cx,x , x ∈
S} ∪ {c[xy,V ],x , x, y ∈ V ⊆ S}.
By applying Theorem 9 we have the following result:
Proposition 16 For every Π ∈ C n , and for every c ∈ M edC (Π), c ≤ c0Π .
According to Remark 9 it would be also possible to obtain a lower bound for the medians
of Π in C.
29
6
Conclusion
In this paper we have shown how to apply the general results of the latticial consensus
theory to the consensus problems for classes of choice functions which are lattices, and in
particular for the choice functions satisfying the heritage, concordance or outcast axioms.
This approach need to study the structural properties of such lattices, what has been done
in Sections 3 and 4.
Some results obtained by this approach have been given in Section 5. One can compare these
results with other results obtained by the more classical approach followed by the russian
school and summarized in Aizerman and Aleskerov’s book ([1]). In fact, this comparison
can concern only the axiomatic approach since the metric approach is not considered (at
least) in this book9 . The main difference is that our axioms on consensus functions are
defined with respect to the irreducible elements of the lattice of choice functions considered,
whereas their axioms are always defined with respect to the choice functions c A,x . These
choice functions are not explicitly considered in their work, but all their consensus functions
satisfy what they call the local property, namely:
For all Π, Π0 , for every A ⊆ S and for every x ∈ A [{i ∈ N : x ∈ ci (A)} = {i ∈ N :
x ∈ c0i (A)}] =⇒ [x ∈ c(A) ⇐⇒ x ∈ c0 (A)].
Since JS = JC = {cA,x , A ⊆ S, x ∈ A} this locality axiom is our J-decisiveness axiom for
the lattices S and C. Aizerman and Aleskerov define also a neutrality and monotonicity
axiom with respect to the choice functions cA,x . For the lattice S (or C) the combination of
their two axioms is equivalent to our neutrality monotonicity axiom (see [21]).
Then for these two lattices we have the same axioms and for instance our Theorem 7 for
choice functions satisfying the concordance axiom (C) corresponds to case 2 of Theorem 6.10
in [1]. For other lattices of choice functions the axioms are different but can lead to the same
classes of consensus functions (in fact these classes are always the class of all federation consensus functions or the restricted classes of oligarchic or co-oligarchic consensus functions).
It is for instance the case for our Theorem 8 concerning the choice functions satisfying the
outcast axiom (O). We obtain the class of join-projection (co-oligarchic) consensus functions
like in Theorem 6.10 quoted above.
Aizerman and Aleskerov’s book contains also many results for various strengthenings of their
basic axiomatic and/or for other classes of choice functions in particular those which are
proper (i.e. such that the choice is never empty) and those obtained by combining the three
fundamental axioms (H), (C) and (O). Recall that one obtains the class of rationalizable (respectively rationalizable by a partial order, or path-independent) choice functions by taking
the choice functions satisfying axioms (H) and (C) (respectively axioms (H), (C) and (O),
or axioms (H) and (O)). These three significant classes of choice functions are not lattices
and so one cannot use Theorem 4 or 5 of our paper. However they are join semilattices with
the set union as join operation. For instance, the ∪-semilattice of path independent choice
functions has been studied in [24] (see also [9], [10]). And on the other hand many results
of axiomatic and metric latticial consensus theories have been extended to semilattices (see
e.g. [11], [12], [13], [14], [18], [19], [21]). Then such results can be applied to the above
semilatticial classes of choice functions, modulo the study of the structural properties of
such semilattices. Here also one can get again some results of Aizerman and Aleskerov’s
book or to find different results. Since our aim was more to illustrate the approach using
the latticial consensus theory than to enumerate results obtained by this approach we send
9
The metric approach has been considered by people like Mirkin ([20]) but not for choice
functions.
30
back to [27] for cases concerning semilattices of choice functions. Note that work remains to
do in this direction since the structure of such semilattices has not always been completely
determined. In fact even in the case of the lattices considered here, we don’t know all the
join-irreducible elements of O as well as the dependence relations β in C and δ and β in O.
We hope to progress on those points in future.
Acknowledgements
The authors thank two anonymous referee for several useful remarks or corrections on the
first version of the paper.
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